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Average value of a function worksheets Grade 12

Find the average value of a function on an interval.

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How to do it

The average value of f from a to b is the integral from a to b divided by (b − a).

the average value of f(x)=2x2+x+7 from x=1 to x=4:472\text{the average value of }f(x) = 2x^{2} + x + 7\ \text{from }x = 1\ \text{to }x = 4:\quad \dfrac{47}{2}
14−1∫14(2x2+x+7) dx\dfrac{1}{4 - 1} \int_{1}^{4} (2x^{2} + x + 7)\, dxthe integral divided by the width
F(x)=2x33+x22+7xF(x) = \dfrac{2x^{3}}{3} + \dfrac{x^{2}}{2} + 7x
(2363)−(496)=1412(\dfrac{236}{3}) - (\dfrac{49}{6}) = \dfrac{141}{2}the integral
14123=472\dfrac{\dfrac{141}{2}}{3} = \dfrac{47}{2}divide by 3

Example problems

  1. the average value of f(x)=2x2+6x+1 from x=2 to x=4:\text{the average value of }f(x) = 2x^{2} + 6x + 1\ \text{from }x = 2\ \text{to }x = 4:\quad
  2. the average value of f(x)=3x2−5x−5 from x=0 to x=4:\text{the average value of }f(x) = 3x^{2} - 5x - 5\ \text{from }x = 0\ \text{to }x = 4:\quad
  3. the average value of f(x)=2x2−6x+2 from x=0 to x=5:\text{the average value of }f(x) = 2x^{2} - 6x + 2\ \text{from }x = 0\ \text{to }x = 5:\quad
  4. the average value of f(x)=3x2+4x+3 from x=1 to x=5:\text{the average value of }f(x) = 3x^{2} + 4x + 3\ \text{from }x = 1\ \text{to }x = 5:\quad

Integrals

  1. Antiderivatives with the power rule
  2. Integrals of 1/x, eˣ, sin and cos
  3. Integration by u-substitution
  4. Evaluating definite integrals
  5. Area under a curve
  6. Area between two curves
  7. Average value of a function